arXiv subjects

Gabriel Claret

Publications and source records attributed to Gabriel Claret.

6 recordsLinked to original sources

arXiv2026

Integral Equation Methods for Scattering by Multifractal Obstacles

Caetano et al. (Proc. R. Soc. A. 481:20230650, 2025) have proposed a formulation for sound-soft acoustic scattering by a compact scatterer O $\subset$ Rn, in which the scattered field is represented as an acoustic Newtonian potential whose density is the solution of an operator equation on a compact set $\Gamma$ $\subset$ O. In the case that $\Gamma$ is Ahlfors-David d-regular (a d-set), for some d $\in$ (n--2, n], they show, moreover, that the operator equation can be interpreted as an integral equation, the integration with respect to d-dimensional Hausdorff measure, and present a convergent Galerkin scheme for numerical computation. In this paper we make a substantial extension of these results so that they apply to more realistic fractal scatterers that are multifractal, in the sense that they have spatially varying fractal dimension. Firstly, we provide, inspired by Claret et al. (J. Math. Pures Appl. 212:103888, 2026), an interpretation of this operator equation as an equation between a trace space on $\Gamma$ and its dual, and, in many cases, relate the density to a notion of the normal derivative of the scattered field on $\Gamma$. Secondly, we show that the operator equation is equivalent to an integral equation on $\Gamma$ whenever $\Gamma$ is the support of a Radon measure $\mu$ such that: (i) the trace operator from H1(Rn) to L2($\Gamma$, $\mu$) is continuous and; (ii) certain canonical singular integrals with respect to $\mu$ are finite; and we characterise a large class of measures for which (i) and (ii) hold. Finally, we show that Galerkin methods based on finite element subspaces of L2($\Gamma$, $\mu$) are convergent if and only if, additionally, C$\infty$\_0 (Rn\$\Gamma$) is dense in the kernel of the trace operator. These results apply, in particular, if $\Gamma$ is a finite union of d-sets with different values of d. In the case that each d-set is the attractor of an iterated function system of contracting similarities, we establish rates of convergence for the Galerkin method.

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arXiv2026

Helmholtz transmission problem and intrinsic impedance scattering problem on extension domains

We consider a transmission problem for the Helmholtz equation with a fixed, positive wavenumber across the boundary of an extension domain. Such a boundary can be Lipschitz, fractal, or of varying Hausdorff dimension. We generalise the notions of layer potential and Neumann-Poincar{\'e} operators, and of Calder{\'o}n projectors in that context. Those boundary operators allow to connect the transmission problem (on the whole space) to one-sided problems -- notably, scattering problems -- with Dirichlet, Neumann and Robin boundary conditions, and restate their well-posedness as boundary equations. Since an extension domain needs no specific boundary measure, the Robin (impedance) condition is not understood in a boundary L^2-type space, rather by duality on the trace space itself. We discuss the well-posedness of the impedance scattering problem in that framework and compare it to the classical L^2 setting. Our analysis allows to generalise optimisation results for acoustic scattering when the obstacle is an extension domain in any dimension.

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arXiv2025

Poincar{\'e}-Steklov operator and Calder{\'o}n's problem on extension domains

We consider Calder{\'o}n's problem on a class of Sobolev extension domains containing non-Lipschitz and fractal shapes. We generalize the notion of Poincar{\'e}-Steklov (Dirichlet-to-Neumann) operator for the conductivity problem on such domains. From there, we prove the stability of the direct problem for bounded conductivities continuous near the boundary. Then, we turn to the inverse problem and prove its stability at the boundary for Lipschitz conductivities, which we use to identify such conductivities on the domain from the knowledge of the Poincar{\'e}-Steklov operator. Finally, we prove the stability of the inverse problem on the domain for W^{2,$\infty$} conductivities constant near the boundary. The last two results are valid in dimension n $\ge$ 3.

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arXiv2024

Convergence of layer potentials and Riemann-Hilbert problem on extension domains

We prove the convergence of layer potential operators for the harmonic transmission problem over a sequence of converging two-sided extension domains. Consequently, the Neumann-Poincar{\'e} operators, Calder{\'o}n projectors, and associated Neumann series converge in this setting. As a result, we generalize the notion of Cauchy integrals and, in a sense, of Hilbert transforms for a class of extension domains. Our approach relies on dyadic approximations of arbitrary open sets, considering convergence in terms of characteristic functions, Hausdorff distance, and compact sets.

math.AP↗

arXiv2024

Layer potential operators for transmission problems on extension domains

We use the well-posedness of transmission problems on classes of two-sided Sobolev extension domains to give variational definitions for (boundary) layer potential operators and Neumann-Poincar{\'e} operators. These classes of domains contain Lipschitz domains, and also domains with fractal boundaries. Although our variational formulation does not involve any measures on the boundary, we recover the classical results in smooth domains by considering the surface measure on the boundary. We discuss properties of these operators and generalize basic results in imaging beyond the Lipschitz case.

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arXiv2023

Existence of optimal shapes for heat diffusions across irregular interfaces

We consider a heat transmission problem across an irregular interface -- that is, non-Lipschitz or fractal -- between two media (a hot one and a cold one). The interface is modelled as the support of a d-upper regular measure. We introduce the proprieties of the interior and exterior trace operators for two-sided extension domains, which allow to prove the well-posedness (in the sense of Hadamard) of the problem on a large class of domains, which contains regular domains, but also domains with variable boundary dimension. Then, we prove the convergence in the sense of Mosco of the energy form connected to the heat content of one of the domains and the heat transfer for ($ε$, $\infty$)-domains. Finally, we prove the existence of an optimal shape maximizing the heat energy transfer in a class of ($ε$, $\infty$)-domains, allowing fractal boundaries, while that optimum can generally not be reached in the class of Lipschitz domains.

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